In a 12-inch skillet over medium heat, add the butter and oil and let the butter melt.
Add the shallots and cook, stirring often until they soften and start to turn a little golden--about 5 minutes.
Add the mustard and vinegar and use a whisk to combine it. Stir vigorously for about 1 minute to help it come together.
Add the green beans and increase the heat to medium high. Toss the green beans to coat in the sauce and add the salt and pepper. You can add half the salt if you'd prefer and the rest at the end if you need it.
Cook until the green beans are hot all the way through and tender--about 15 minutes, stirring frequently.
Serve warm with the sauce poured over the top and add some toasted almonds or walnuts on top if desired.